The Riemannian wave operator is a globally hyperbolic
linear partial differential operator of real principal
type. Therefore, the Riemannian distance function and the propagation
of a singularity initiated by a point source in space time are related
to one another.
We let \(u\)  be the solution of the Riemannian wave equation as in (REF ).
In [1]}, [2]} it is shown
that the image, \(\Lambda \) , of the wave front set of \(u\) , under the
musical isomorphism \(T^\ast M \ni (x,\xi ) \mapsto (x,g^{ij}(x)\xi _i) \in TM\) , coincides
with the image of the unit sphere \(S_{p}M\)  at \(p \in M^{int}\)  under the geodesic flow of \(g\) . Thus \(\Lambda \cap \partial (S M)\) , where \(SM\)  is the unit sphere bundle of \((M,g)\) , coincides with the exit
directions of geodesics emitted from \(p\) .
In [3]} the authors show that if \((M,g)\)  is a
compact smooth non-trapping Riemannian manifold with smooth strictly
convex boundary, then generically the scattering data of point
sources \((\partial M, R_{\partial M}(M))\)  determine \((M,g)\)  up to an isometry.
Here, \(R_{\partial M}(p) \in R_{\partial M}(M)\)  for \(p \in M\)  stands for
the collection of tangential components to boundary of exit directions
of geodesics from \(p\)  to \(\partial M\) .
